Issue: 2024/Vol.34/No.2, Pages 65-84
ON THE NASH EQUILIBRIA OF A SIMPLE DISCOUNTED DUEL
Athanasios Kehagias
Cite as: A. Kehagias. On the Nash equilibria of a simple discounted duel. Operations Research and Decisions 2024: 34(2), 65-84. DOI 10.37190/ord240205
Abstract
We formulate and study a two-player, duel game as a nonzero-sum discounted stochastic game. Players P1, and P2 are standing in place and, in each turn, one or both may shoot at the other player. If Pn shoots at Pm (m ≠ n), either he hits and kills him (with probability pn) or he misses him and Pm is unaffected (with probability 1 − pn). The process continues until at least one player dies; if nobody ever dies, the game lasts an infinite number of turns. Each player receives a unit payoff for each turn in which he remains alive; no payoff is assigned to killing the opponent. We show that the always-shooting strategy is a NE but, in addition, the game also possesses so-called cooperative (i.e., non-shooting) Nash equilibria in both stationary and nonstationary strategies. A certain similarity to the repeated Prisoner’s Dilemma is also noted and discussed.
Keywords: game theory, Nash equilibrium, duel
Received: 8 May 2023 Accepted: 12 April 2024
Published online: 8 July 2024